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Properties

Label 16.0.65859007567...6441.1
Degree $16$
Signature $[0, 8]$
Discriminant $19^{12}\cdot 29^{14}$
Root discriminant $173.25$
Ramified primes $19, 29$
Class number $1625$ (GRH)
Class group $[5, 5, 65]$ (GRH)
Galois group $C_8: C_2$ (as 16T6)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![214651801, 0, 300750338, 0, 91602093, 0, 9841099, 0, 175088, 0, -28619, 0, -1024, 0, 24, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 + 24*x^14 - 1024*x^12 - 28619*x^10 + 175088*x^8 + 9841099*x^6 + 91602093*x^4 + 300750338*x^2 + 214651801)
 
gp: K = bnfinit(x^16 + 24*x^14 - 1024*x^12 - 28619*x^10 + 175088*x^8 + 9841099*x^6 + 91602093*x^4 + 300750338*x^2 + 214651801, 1)
 

Normalized defining polynomial

\( x^{16} + 24 x^{14} - 1024 x^{12} - 28619 x^{10} + 175088 x^{8} + 9841099 x^{6} + 91602093 x^{4} + 300750338 x^{2} + 214651801 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 8]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(658590075672306993006756541342366441=19^{12}\cdot 29^{14}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $173.25$
magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
 
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
Ramified primes:  $19, 29$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois over $\Q$.
This is not a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{7} a^{7} - \frac{1}{7} a$, $\frac{1}{14} a^{8} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{14} a^{2} - \frac{1}{2}$, $\frac{1}{14} a^{9} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{14} a^{3} - \frac{1}{2} a$, $\frac{1}{14} a^{10} - \frac{1}{14} a^{7} - \frac{1}{2} a^{5} - \frac{1}{14} a^{4} - \frac{1}{2} a^{2} - \frac{3}{7} a$, $\frac{1}{14} a^{11} - \frac{1}{2} a^{6} + \frac{3}{7} a^{5} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{98} a^{12} - \frac{3}{98} a^{10} + \frac{1}{49} a^{8} - \frac{11}{49} a^{6} - \frac{1}{2} a^{5} + \frac{17}{98} a^{4} - \frac{1}{2} a^{3} + \frac{5}{98} a^{2} - \frac{1}{2} a$, $\frac{1}{2254} a^{13} - \frac{59}{2254} a^{11} + \frac{9}{2254} a^{9} - \frac{32}{1127} a^{7} + \frac{563}{2254} a^{5} + \frac{48}{1127} a^{3} - \frac{1}{2} a^{2} + \frac{83}{322} a$, $\frac{1}{3502101580557429911642} a^{14} + \frac{5459186138863131979}{1751050790278714955821} a^{12} - \frac{19935707660289618677}{3502101580557429911642} a^{10} + \frac{68448115542878280919}{3502101580557429911642} a^{8} - \frac{1}{14} a^{7} - \frac{345041520928359494525}{3502101580557429911642} a^{6} - \frac{1}{2} a^{5} + \frac{188033785599401050696}{1751050790278714955821} a^{4} - \frac{1}{2} a^{3} + \frac{590077223807905348051}{1751050790278714955821} a^{2} + \frac{1}{14} a + \frac{785242415116775203}{3107454818595767446}$, $\frac{1}{318691243830726121959422} a^{15} - \frac{13185542772711472697}{159345621915363060979711} a^{13} - \frac{2036805259259234125992}{159345621915363060979711} a^{11} + \frac{4244048473264055087928}{159345621915363060979711} a^{9} - \frac{5463019607155588478087}{318691243830726121959422} a^{7} - \frac{1}{2} a^{6} - \frac{87908277552516248923191}{318691243830726121959422} a^{5} + \frac{79348519601117631266921}{159345621915363060979711} a^{3} - \frac{1}{2} a^{2} - \frac{117598496083934608590}{3251951467660470632239} a$

magma: IntegralBasis(K);
 
sage: K.integral_basis()
 
gp: K.zk
 

Class group and class number

$C_{5}\times C_{5}\times C_{65}$, which has order $1625$ (assuming GRH)

magma: ClassGroup(K);
 
sage: K.class_group().invariants()
 
gp: K.clgp
 

Unit group

magma: UK, f := UnitGroup(K);
 
sage: UK = K.unit_group()
 
Rank:  $7$
magma: UnitRank(K);
 
sage: UK.rank()
 
gp: K.fu
 
Torsion generator:  \( -1 \) (order $2$)
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
Fundamental units:  Units are too long to display, but can be downloaded with other data for this field from 'Stored data to gp' link to the right (assuming GRH)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 1385349207.51 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$OD_{16}$ (as 16T6):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 16
The 10 conjugacy class representatives for $C_8: C_2$
Character table for $C_8: C_2$

Intermediate fields

\(\Q(\sqrt{-551}) \), \(\Q(\sqrt{29}) \), \(\Q(\sqrt{-19}) \), \(\Q(\sqrt{-19}, \sqrt{29})\), 4.4.8804429.1, 4.0.24389.1, 8.0.77517970016041.1, 8.4.811535628097933229.1 x2

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

Degree 8 sibling: data not computed

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/5.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/7.1.0.1}{1} }^{16}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/23.1.0.1}{1} }^{16}$ R ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.

magma: p := 7; // to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
magma: idealfactors := Factorization(p*Integers(K)); // get the data
 
magma: [<primefactor[2], Valuation(Norm(primefactor[1]), p)> : primefactor in idealfactors];
 
sage: p = 7; # to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
sage: [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]
 
gp: p = 7; \\ to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
gp: idealfactors = idealprimedec(K, p); \\ get the data
 
gp: vector(length(idealfactors), j, [idealfactors[j][3], idealfactors[j][4]])
 

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
19Data not computed
29Data not computed